Abstract
We study a defocusing semilinear wave equation, with a power nonlinearity \(|u|^{p-1}u\), defined outside the unit ball of \(\mathbb {R}^{n}\), \(n\ge 3\), with Dirichlet boundary conditions. We prove that if \(p>n+3\) and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a Strichartz estimate for the exterior linear wave equation perturbed with a large, time dependent potential.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Consider the Cauchy problem for the defocusing wave equation on \(\mathbb {R}_{t}\times \mathbb {R}^{n}_{x}\):
with Sobolev initial data \((u_{0},u_{2})\in H^{s}\times H^{s-1}\). The existence of global solutions to this problem has been explored in considerable detail. The energy critical power for global smooth solvability is \(p_{cr}(n)=1+\frac{4}{n-2}\) for \(n\ge 3\), while \(p_{cr}(1)=p_{cr}(2)=\infty \). Global existence in the energy space is known for \(p\le p_{cr}(n)\) [4, 5, 8]; regularity for \(p=p_{cr}\) has been explicitly proved up to dimension 7 [5, 7] and should hold for all dimensions. For supercritical nonlinearities \(p>p_{cr}(n)\), very little is known, and the question of global existence or blow up is still open.
If we restrict to spherically symmetric solutions, the difficulty of the problem is essentially the same. However, if a radial solution blows up, the blow up must occur at the origin. This follows at once from the Sobolev embedding for radial functions
which is a well known special case of the family of inequalities
(see [3]). Here the norm \(L^{p}_{|x|}L^{r}_{\omega }\) is an \(L^{p}\) norm in the radial direction of the \(L^{r}_{\omega }\) norm in the angular direction. We give a precise statement of this blow–up alternative result including its short proof:
Proposition 1.1
Let \(u_{0},u_{1}\) be spherically symmetric test functions, and T the supremum of times \(\tau >0\) such that a smooth solution u(t, x) of (1.1) exists on \([0,\tau ]\times \mathbb {R}^{n}\). Assume \(T<\infty \). Then u blows up at \(x=0\) as \(t\rightarrow T\), in the sense that for any \(R>0\), u is unbounded on the cylinder \(\{|x|<R,\ t<T\}\).
Proof
By standard local existence results (e.g. [9], Chapter 5) we know that \(T>0\), and the solution is spherically symmetric for all \(t\in [0,T)\) by uniqueness of smooth solutions. Denote the energy of the solution u at time t by
By conservation of energy and the radial estimate (1.2) we have
Assume by contradiction that \(|u(t,x)|\le M\) for (t, x) in some cylinder \(\{|x|<R,\ t<T\}\), then by the previous bound we get
and by standard arguments we can extend u to a strip \([0,T']\times \mathbb {R}^{n}\) for some \(T'>T\), against the assumptions. \(\square \)
In order to obtain a global radial solution, it would be sufficient to prevent blow-up at 0. This is the case if we replace the whole space with the exterior of a ball:
and consider the mixed problem on \(\mathbb {R}^{+}\times \Omega \) with Dirichlet boundary conditions
In this situation the local solution can not blow up, due to the conservation of \(H^{1}\) energy and the radial estimate (1.2). Note indeed that estimate (1.2) is valid also for functions in \(H^{1}_{0}(\Omega )\), since it holds on the dense subspace \(C_{c}^{\infty }(\Omega )\). This implies the existence of a global radial solution for arbitrary powers \(p>1\). Indeed, we have the following result, where we use the notation
Proposition 1.2
Let \(\Omega =\mathbb {R}^{n}\setminus \overline{B(0,1)}\), \(n\ge 2\), \(p>1\) and let \(u_{0}\in H^{1}_{0}(\Omega )\cap H^{2}(\Omega ) \cap L^{p+1}(\Omega )\), \(u_{1}\in H^{1}_{0}(\Omega )\) be two radially symmetric functions.
Then the mixed problem (1.4)
has a global solution \(u\in C^{2}L^{2}\cap C^{1}H^{1}_{0}\cap CH^{2}\), satisfying the conservation of energy
and the uniform bound
If \(v\in C^{2}L^{2}\cap C^{1}H^{1}_{0}\cap CH^{2}\) is a second solution of (1.4) with the same data which is either radially symmetric in x, or locally bounded on \(\mathbb {R}^{+}\times \Omega \), then \(v \equiv u\).
The existence of large radial solutions suggests naturally the question of stability: do nonradial perturbations of radial data give rise to global solutions?
The uniform bound (1.6) is not sufficient for a perturbation argument. However, we can prove an actual decay estimate, obtained by reduction to a mixed problem with moving boundary on \(\mathbb {S}^{n}\) via the Penrose transform. Define for \(M>1\) the quantity
Then we have
Theorem 1.3
(Decay of the radial solution) Let u be the solution constructed in Proposition 1.2. Assume that for some \(M>1\) the data satisfy
Then the following decay estimate holds
If the initial data are smoother then regularity propagates, and in addition higher Sobolev norms remain bounded as \(t\rightarrow \infty \). In order to state the regularity result, we introduce briefly the nonlinear compatibility conditions, discussed in greater detail in Sect. 3 (see Definition 3.3 and also Definition 2.1). For the mixed problem on \(\mathbb {R}^{+}\times \Omega \)
we define formally the sequence of functions \(\psi _{j}\) as follows:
where in the definition of \(\psi _{j}\) we set recursively \(\partial _{t}^{k}u(0,x)=\psi _{k}(x)\) for \(0\le k\le j-2\). Then we say that the data \((u_{0},u_{1},f(s))\) satisfy the nonlinear compatibility conditions of order \(N\ge 1\) if
Note that if f(s) vanishes at \(s=0\) of sufficient order, to satisfy the nonlinear compatibility conditions it is sufficient to assume that the initial data \(u_{0},u_{1}\) belong to \(H^{k}_{0}(\Omega )\) for k large enough.
We can now state the higher regularity result for the radial solutions. In the statement we use the norm \(\Vert v\Vert _{Y^{\infty ,2;k}}= \sum _{|\alpha |\le k} \Vert \partial ^{\alpha }_{t,x}u\Vert _{L^{2}L^{\infty }}\).
Theorem 1.4
Let \(n\ge 3\), \(p>N\ge 1\), \(p> \frac{2n}{n-2}\). Assume the radially symmetric data \(u_{0}\in H^{N+1}(\Omega )\), \(u_{1}\in H^{N}\) and \(f(u)=|u|^{p-1}u\) satisfy the nonlinear compatibility conditions of order N and condition (1.8).
Then the radial solution constructed in Proposition 1.2 belongs to \(C ^{k}H^{N+1-k}\cap C^NH^{1}_{0}\) for all \( 0\le k\le N+1\) and satisfies (besides (1.5)) the uniform bounds
The main result of the paper is the following:
Theorem 1.5
Let \(n\ge 3\), \(N\ge \frac{5}{2}n\), \(p> n+3\), and \(u_{0},u_{1}\) radial functions satisfying the assumptions in Theorem 1.4. Then there exists \(\epsilon =\epsilon (u_{0},u_{1})>0\) such that the following holds.
Assume \(v_{0}\in H^{N+1}(\Omega )\), \(v_{1}\in H^{N}(\Omega )\) and \(f(u)=|u|^{p-1}u\) satisfy the nonlinear compatibility conditions of order N and \(\Vert u_{0}-v_{0}\Vert _{H^{N+1}}< \epsilon \), \(\Vert u_{1}-v_{1}\Vert _{H^{N}}< \epsilon \). Then Problem (1.4) with data \(v_{0},v_{1}\) has a global solution \(v\in C^{k}H^{N-k}\cap C^{N-1}H^{1}_{0}\), \(0\le k\le N+1\).
In other words, nonradial perturbations of large radial initial data in a high Sobolev norm do give rise to global smooth quasiradial solutions. Note that these solutions are unique in the class of locally bounded global solutions, but one can not in principle exclude the existence of other energy class solutions. However, one can prove a weak–strong uniqueness result, which implies in particular that the solution constructed in Theorem 1.5 is the unique energy class solution satisfying an energy inequality:
Theorem 1.6
Suppose all the assumptions in Theorem 1.5 are satisfied. Let I be an open interval containing [0, T] and \(\underline{v}\in C(I;H^{1}_{0})\cap C^{1}(I;L^{2}) \cap L^{\infty }(I;L^{p+1}(\Omega ))\) a distributional solution for \(0\le t\le T\) to Problem (1.4) with the same initial data as v, which satisfies an energy inequality \(E(\underline{v}(t))\le E(\underline{v}(0))\) (see (1.5)). Then we have \(\underline{v}(t)=v(t)\) for \(0\le t\le T\).
Remark 1.1
Similar results can be proved for other dispersive equations, notably for the nonlinear Schrödinger equation. This will be the topic of further work in preparation.
The proof of Theorem 1.5 is based on perturbing the radial solution u(t, x) with a small term w(t, x) satisfying the equation
which can be written in the form
To solve the last equation, we prove an energy–Strichartz estimate for the exterior problem with potential
as an application of the results of [1, 6, 11]; note that this part of the argument holds also on arbitrary non trapping exterior domains. The Strichartz estimate for the wave equation with potential is proved by reduction to the constant coefficient case, which is possible since from the previous results we know that the potential V has rapid decay in (t, x). Finally, Theorem 1.6 is an adaptation of a weak–strong uniqueness result due to M. Struwe [12].
The plan of the paper is the following. The linear theory and the perturbed energy–Strichartz estimates are developed in Sect. 2. In Sect. 3 the global radial solution is constructed and its regularity and decay properties are studied. In the final Sects. 4, 5 we prove the main results Theorems 1.5 and 1.6.
2 Linear theory
Denote by \(-\Delta _{D}\) the Dirichlet Laplacian, that is the nonnegative selfadjoint operator with domain \(H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\), and by \(\Lambda \) its nonnegative selfadjoint square root
We have \(\Lambda ^{2k}=(-\Delta _{D})^{k}\) for integer \(k\ge 0\), and
so that
We shall make repeated use of the equivalence
valid for \(g\in D(\Lambda )=H^{1}_{0}(\Omega )\). The solution of the special mixed problem
with \(u_{1}\in H^{1}_{0}(\Omega )\) can be represented in the form
Then the solution of the full linear mixed problem on \(\mathbb {R}^{+}\times \Omega \)
takes the form
where
We shall be concerned with several global in time estimates of S(t) and of the solution to (2.2). We work for positive times \(t>0\) only, but it is clear that all results are time–reversible. Directly from the spectral theory one gets
As a consequence one gets the basic energy estimate for solutions of (2.2):
valid for all \(T>0\), with a constant independent of T.
Higher regularity results require compatibility conditions. Given the data \((u_{0},u_{1},F)\) we define recursively the sequence of functions \(h_{j}\) as follows:
The function \(h_{j}\) is obtained by applying \(\partial _{t}^{j-2}\) to the equation \(u_{tt}=\Delta u+F\).
Definition 2.1
(Linear compatibility conditions) We say that the data \((u_{0},u_{1},F)\) satisfy the linear compatibility conditions of order \(N\ge 1\) if \((u_{0}, u_{1})\in H^{N+1}(\Omega )\times H^{N}(\Omega )\), \(F\in C^{k}H^{N-k}(\Omega )\) for \(0\le k\le N\), and
To formulate estimates of u in a compact format we introduce a few notations. We write for short for any interval \(I \subseteq \mathbb {R}\) and \(T\ge 0\)
Moreover we denote the \(L^{p}L^{q}\) norm of all spacetime derivatives up to the order N by
When \(I=[0,T]\) or \(I=[0,+\infty )\) we write also
The following result is standard, and valid for general domains \(\Omega \) with sufficiently smooth compact boundary. We use the inequality \(\Vert \Lambda ^{-1}g\Vert _{L^{2}}\lesssim \Vert g\Vert _{L^{\frac{2n}{n+2}}}\) in the formulation of (2.9).
Proposition 2.2
Let \(N\ge 1\), and assume \(u_{0}\in H^{N+1}(\Omega )\), \(u_{1}\in H^{N}(\Omega )\) and \(F\in C^{k}H^{N-k}(\Omega )\) for \(0\le k\le N\) satisfy the linear compatibility conditions of order N. Then Problem (2.2) has a unique solution belonging to
The solution satisfies for all \(T>0\) the energy estimates
with a constant independent of T.
We next recall Strichartz estimates for the exterior problem, following [1, 6, 11]. These estimates are valid on the exterior of any strictly convex obstacle with smooth boundary in \(\mathbb {R}^{n}\), \(n\ge 2\). With our notations one has
provided \(n\ge 3\) and
A couple (q, r) as in (2.12) is called admissible; note that the endpoint \((q,r)=(2,\frac{2(n-1)}{n-3})\) is not included and it is not known if the estimate holds also in this case.
One can further extend the range of indices by combining (2.11) with Sobolev embedding. We shall focus on the following special case:
valid for \(n\ge 3\) and for couples of indices of the form
Then we have:
Proposition 2.3
The solution u to (2.2) saitisfies, for any interval I containing 0, the Sobolev–Strichartz estimate
provided \(n\ge 3\) and (q, r) satisfy (2.14).
Proof
The proof is a standard application of the Christ–Kiselev Lemma to the representation (2.3) of the solution. \(\square \)
Differentiating (2.2) with respect to t and applying the usual recursive procedure one gets, more generally, the following higher order estimates;
Proposition 2.4
Assume the data \((u_{0},u_{1},F)\) of (2.2) satisfy the compatibility conditions of order \(N\ge 1\). Then, for all (q, r) as in (2.14) and for any interval I of length \(|I| \gtrsim 1\) containing 0, the solution satisfies the estimates
Proof
We give a sketch of the proof. Applying \(\partial _{t}\) to the equation, by (2.15) we get
since \(u_{tt}(0)=\Delta u_{0}+F(0,x)\) from the equation. We note that \(\Vert F(0,\cdot )\Vert _{L^{2}}\lesssim \Vert F\Vert _{Y^{1,2;1}_{I}}\) provided the interval has length \(|I|\gtrsim 1\), and this gives the estimate for \(u_{t}\). In a similar way one can estimate all derivatives \(\partial _{t}^{j}u\). We next estimate \(\Delta u=u_{tt}-F\):
The \(u_{tt}\) term has already been estimated. As to the second term, we note that
which is the endpoint \(\delta =0\) in (2.14). Moreover,
and by interpolation and Sobolev embedding
which is the other endpoint \(\delta =1\) in (2.14). This argument can be modified in the case \(n=3\) by using the Sobolev embedding into BMO instead of \(L^{\infty }\). Again by interpolation we get
for all (q, r) as in (2.14), and we conclude
Also by interpolation this covers the case \(N=1\) of (2.16). For larger values of N one proceeds in a similar way by recursion, using the embedding \(Y^{1,2;N}\hookrightarrow Y^{q,r;m-2}\) just proved for all (q, r) as in (2.14). \(\square \)
We shall also need estimates for the exterior wave equation with a time dependent potential. We denote by \(u=S_{V}(t;t_{0})g\) the solution of the mixed problem with initial data at time \(t=t_{0}\)
and by \(u=S_{V}'(t;t_{0})f\) (which is not \(\partial _{t}S_{V}g\)) the solution of
If V is a sufficiently smooth potential with good behaviour at infinity, the existence and uniqueness of a solution is standard. The solution of the full problem
can be represented by Duhamel as
Note also that it is not necessary to modify the compatibility conditions, provided the potential V has a sufficient regularity. Indeed the correct condition would require
but the term \(\partial _{t}^{j-2-\ell }V(0,x)h_{\ell }\) is already in \(H^{1}_{0}\) since \(h_{\ell }\in H^{1}_{0}\), and can be omitted. By Duhamel we can write \(S_{V}(t;s)\), \(S_{V}'(t,s)\) as perturbations of S(t), \(\partial _{t}S(t)\):
Proposition 2.5
(Perturbed energy–Strichartz estimate) Let \(n\ge 3\), \(m\ge 1\). Assume the data \((u_{0},u_{1},F)\) satisfy the compatibility conditions of order m, and that
Then for any interval I containing \(t_{0}\) the solution of Problem (2.17) satisfies
provided the couple (q, r) is of the form (2.14).
Proof
We can assume \(I=I(t)=[t_{0},t]\); the proof for \(t<t_{0}\) is identical. Since u solves \(\square u=F-Vu\), by (2.9) we get
Noting that
and \(a(s)=\Vert V(s)\Vert _{L^{n}}\) is integrable by (2.21), by Gronwall’s Lemma we deduce (2.22).
In a similar way, by (2.10) and (2.16) we can write
We have
where
is integrable on \(\mathbb {R}\) by (2.21). Using again Gronwall’s inequality we obtain (2.23). \(\square \)
3 The global radial solution
This section is devoted to the proof of Proposition 1.2 and Theorems 1.3, 1.4. We begin with a few preliminary results on the mixed problem
For data of low regularity, a solution to (3.1) is intended to be a solution of the integral equation
with S(t) as in (2.1). We will give only sketchy proofs of standard results, which are virtually identical to the corresponding ones for semilinear wave equations on \(\mathbb {R}^{n}\) (for which we refer e.g. to Chapter 6 of [9]).
Lemma 3.1
Assume \(f:\mathbb {R}\rightarrow \mathbb {R}\) is (globally) Lipschitz with \(f(0)=0\). Then for any initial data \((u_{0}, u_{1})\in H^{1}_{0}\times L^{2}(\Omega ) \) Problem (3.1) has a unique solution u(t, x) in \(C H^{1}_{0}(\Omega )\cap C^{1}L^{2}(\Omega )\). The solution satisfies the energy bound for all \(t>0\)
Proof
Apply a contraction argument in the space \(C([0,T]; H^{1}_{0}(\Omega )) \cap C^{1}([0,T];L^{2}(\Omega ))\) to (3.2), with \(T>0\) sufficiently small, using the energy estimates (2.4), (2.5). The lifespan T depends only on the \(H^{1}\times L^{2}\) norm of the data, thus we can iterate to a global solution. Estimate (3.3) is a byproduct of the proof. \(\square \)
Lemma 3.2
Consider the solution u constructed in Lemma 3.1. Assume in addition that \(f\in C^{2}\), \(u_{0}\in H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\), \(u_{1}\in H^{1}_{0}(\Omega )\). Then u belongs to \(C^{2}L^{2}(\Omega )\cap C^{1}H^{1}_{0}(\Omega )\cap C H^{2}(\Omega )\) and solves (3.1) in both distributional and a.e. sense.
If we further assume that \(0\le f(s)s\lesssim F(s)\) for \(s\in \mathbb {R}\), where \(F(s)=\int _{0}^{s}f(\sigma )d\sigma \), then the solution satisfies for all times the energy identity
Proof
Differentiate the equation once w.r.to time and call formally \(v=u_{t}\). Then v must solve
where we may regard \(f'(u)\) as an \(L^{\infty }\) coefficient. As data we take \(v(0)=u_{t}\in H^{1}_{0}(\Omega )\) and \(v_{t}(0)=u_{tt}(0)=\Delta u(0)-f'(u(0))v(0)\in L^{2}(\Omega )\). By the previous result, this problem has a global solution v(t, x), satisfying an energy estimate like u. Since \(u(0)+\int _{0}^{t}v\) solves the original equation (3.1), by uniqueness for (3.1) we conclude that \(v=u_{t}\) and hence u has the claimed regularity. The energy conservation follows by direct computation. \(\square \)
Note that the assumption \(0\le sf(s)\lesssim F(s)\) is sufficient to prove the existence of a global weak solution for data in \(H^{1}_{0}(\Omega )\times L^{2}(\Omega )\), even if f is not Lipschitz (Segal’s Theorem). This can be proved like in the case of the whole space \(\mathbb {R}^{n}\) by approximating f with a sequence of truncated Lipschitz functions and using weak compactness. The weak solution thus constructed satisfies then a weaker energy inequality \(E(t)\le E(0)\) (proved using Fatou’s Lemma). We shall not need this variant in the sequel.
For smoother data, one can prove a local existence theorem which does not require a global Lipschitz condition, similarly to the case \(\Omega =\mathbb {R}^{n}\). However, one must assume suitable compatibility conditions, analogous to the linear ones from Definition 2.1. Define formally a sequence of functions \(\psi _{j}\), \(j\ge 0\) as follows: differentiating the equation \(u_{tt}=\Delta u+f(u)\) with respect to time, set
where values of \(\partial _{t}^{k}u(0,x)\) for \(0\le k\le j-2\), required to compute \(\psi _{j}\), are set recursively equal to \(\psi _{k}\). For instance,
and so on. Then we have:
Definition 3.3
(Nonlinear compatibility conditions) We say that the data \((u_{0},u_{1},f)\) satisfy the nonlinear compatibility conditions of order \(N\ge 1\) if \(~(u_{0},u_{1})\in H^{N+1}(\Omega )\times H^{N}(\Omega )\), \(f\in C^{N-2}(\mathbb {R};\mathbb {R})\) and we have
Remark 3.1
Conditions (3.5) are implied by a number of simpler assumptions on the data. For instance, if \(f\in C^{N-2}\) and one assumes
then one checks easily that \((u_{0},u_{1},f)\) satisfiy the nonlinear compatibility conditions of order N.
Lemma 3.4
(Local existence) Let \(N> \frac{n}{2} \), \((u_{0},u_{1})\in H^{N+1}(\Omega )\times H^{N}(\Omega )\), \(f\in C^{N}\) and assume \((u_{0},u_{1},f)\) satisfy the nonlinear compatibility conditions of order N. Then there exists \(0<T\le +\infty \), depending only on the \(H^{N+1}\times H^{N}\) norm of \((u_{0},u_{1})\), such that Problem (3.1) has a unique solution \(u\in C^{k}([0,T);H^{N+1-k}(\Omega ))\), \(0\le k\le N+1\). The solution belongs to \(C^{N}([0,T);H^{1}_{0}(\Omega ))\).
Moreover, if \(T^{*}\) is the maximal time of existence of such a solution, then either \(T^{*}=+\infty \) or \(\Vert u(t,\cdot )\Vert _{L^{\infty }}\rightarrow \infty \) as \(t \uparrow T^{*}.\)
Proof
The existence part is completely standard; it is usually proved for more general quasilinear equations, which require higher smoothness of the data; see e.g. Theorem 3.5 in [10], where local existence is proved for a nonlinear term of the form \(f(t,x,\partial _{t}^{j}\partial _{x}^{\alpha }u)\) with \(j+|\alpha |\le 2\), \(j\le 1\) (and a regularity of order \(\lfloor \frac{n}{2} \rfloor +8\) is imposed on the data). The proof is based on a contraction mapping argument, combined with Moser type estimates of the nonlinear term. The final blow up alternative in the statement is a byproduct of the proof. \(\square \)
3.1 Proof of Proposition 1.2
Fix \(M>0\) and define \(f_{M}(s)=\min \{|s|,M\}^{p-1}s\). Then \(f_{M}\) is Lipschitz and the problem
has a global, unique, radially symmetric solution \(u\in C^{2}L^{2}(\Omega )\cap C^{1}H^{1}_{0}(\Omega )\cap C H^{2}(\Omega )\) satisfying the bound (3.4) with \(F=F_{M}=\int _{0}^{s}f_{M}\). Combining (3.4) with (1.2) we get
for some universal constant \(C_{0}\). Since \(|x|\ge 1\) on \(\Omega \), this gives
and if we choose \(M=C_{0}K+1\) we see that \(f_{M}(u(t,x))=f(u(t,x))\), i.e., u(t, x) is a global solution of the untruncated problem
The same argument guarantees also uniqueness of radially symmetric solutions. More generally, two solutions in \(H^{2}\cap L^{\infty }\) with the same data must coincide, as an elementary argument based on classical energy estimates shows. Since energy estimates are valid also on domains of dependence, by finite speed of propagation, the same arguent shows uniqueness of solutions in \(H^{2}\cap L^{\infty }_{loc}\). In particular, a locally bounded solution with radial data must coincide with the radial solution. The proof of Proposition 1.2 is concluded.
3.2 Proof of Theorem 1.3
We now prove the decay estimate (1.9), using the Penrose transform. We recall its definition. Describe the sphere \(\mathbb {S}^{n}\) using coordinates \((\alpha ,\theta )\) with \(\alpha \in (0,\pi )\) and \(\theta \in \mathbb {S}^{n-1}\), as \(\mathbb {S}^{n}_{\alpha ,\theta } =(0,\pi )_{\alpha }\times \mathbb {S}^{n-1}_{\theta }\). Denoting with \(d \theta _{\mathbb {S}^{n-1}}^{2}\) the metric of \(\mathbb {S}^{n-1}_{\theta }\), the metric on \(\mathbb {S}^{n}\) can then be written as
Similarly, on \(\mathbb {R}^{n}_{x}=\mathbb {R}^{+}_{r} \times \mathbb {S}^{n-1}_{\theta }\), use polar coordinates \((r,\theta )\) with \(r\in (0,+\infty )\) and \(\theta \in \mathbb {S}^{n-1}\), so that the euclidean metric can be written \(dr^{2}+r^{2}d \theta _{\mathbb {S}^{n-1}}^{2}\). Then we can define the Penrose map \(\Pi : \mathbb {R}_{t}\times \mathbb {R}_{x}\rightarrow \mathbb {R}_{T}\times \mathbb {S}^{n}\) as
where
The map \((t,r)\mapsto (T,\alpha )\) takes the quadrant
to the triangle
so that \(\Pi \) maps \(\mathbb {R}^{+}\times \mathbb {R}^{n}\) to the positive half of the Einstein diamond
The boundary \(|x|=1\) i.e. \(r=1\) is mapped by \(\Pi \) to a curve described parametrically by
for \(t\ge 0\). We denote this curve by
(i.e., \(\gamma =\Gamma ^{-1}\)). One has the explicit (but not particularly useful) formulas
It is easy to check that
We denote by \(\omega \) the conformal factor
Note that \(\omega >0\) on \(\mathbb {E}^{+}\). The inverse of \(\Pi \) (defined on \(\mathbb {E}^{+}\)) can be written
We define a new function \(U(T,\alpha ,\theta )\) via
Since u, U are independent of \(\theta \), we shall write simply \(U(T,\alpha )\). Commuting \(\square \) with \(\Pi \) gives
so that U is a solution of the equation
on the subset \(\mathbb {E}_{\Omega }\) of \(\mathbb {R}\times \mathbb {S}^{n}\) given by the conditions
which is the image of \(\mathbb {R}^{+}_{t}\times \Omega \) via \(\Pi \). We introduce also the notation
for the slice of \(\Pi (\mathbb {R}^{+}\times \Omega )\) at time T. Note that in coordinates, Eq. (3.13) reads
We plan to extend the solution beyond the line \(T+\alpha =\pi \), i.e., in the region where \(\omega <0\). Thus we consider the following extended equation on \((T,\alpha )\in [0,\pi ]^{2}\):
where we have replaced \(\omega \) by
The solution U satisfies the identity
We can now extend U to a larger domain in the cylinder \(\mathbb {R}_{T}\times \mathbb {S}^{n}\). Recall that the data \(u_{0},u_{1}\) satisfy \(C_{M}(u_{0},u_{1})<\infty \) with \(C_{M}(u_{0},u_{1})\) as in (1.7). Thus if we fix a smooth cutoff function \(\chi (x)\) equal to 0 for \(|x|\le M+1\) and equal to 1 for \(|x|\ge M+2\), we have
Denote by \(\widetilde{U}_{0},\widetilde{U}_{1}\) the functions obtained by applying the transformation (3.12) to \(\chi u_{0}, \chi u_{1}\) respectively (with \(t=0\)). By the first Lemma in Section 4 of [2] we have then
In order to solve (3.13) locally via the energy method we require that the coefficient \(\widetilde{\omega }^{\nu }\) be sufficiently smooth i.e. \(\widetilde{\omega }^{\nu }\in C^{N_{0}}\). This is true as soon as
Then a standard local existence result guarantees the existence of a local solution \(\widetilde{U}\) to Eq. (3.13) with data \(\widetilde{U}_{0},\widetilde{U}_{1}\) on some strip \([0,\delta )\times \mathbb {S}^{n}\). The lifespan \(\delta \), which can be assumed \(\ll 1\), depends only on
where \(C_{M}(u_{0},u_{1})\) was defined in (1.7). Comparing \(\widetilde{U}\) with the solution U constructed above, and noting that Eq. (3.13) has finite speed of propagation equal to 1, by local uniqueness we see that \(U, \widetilde{U}\) must coincide on the forward dependence domain emanating from the set
Thus we can glue the two solutions, at least for \(0<~T<\delta /2\), and we obtain an extended solution of (3.13), which we denote again \(U(T,\alpha )\), defined on the larger domain
We next prove that the energy of U, defined as
remains bounded for \(0\le T<\delta /2\). To this end we integrate identity (3.15) on a slice
for arbitrary times \(0\le T_{1}\le T_{2}<\delta /2\). Dropping negative terms at the RHS, we are left with the inequality
where \(\nu _{\alpha },\nu _{T}\) are the components of the exterior normal to the curve \(T=\gamma (\alpha )\), i.e.,
The Dirichlet condition \(U(\gamma (\alpha ),\alpha )=0\) along the curve implies
Thus the RHS of (3.17) is equal to
Recalling (3.11), we see that the RHS of (3.17) is negative, and we conclude that the energy is nonincreasing as claimed:
Now, consider the set, which is a dependence domain for (3.13) (keep in mind that the speed of propagation for (3.13) is exactly 1):
We have already extended U to the part of \(\mathbb {D}\) in the time strip \(\delta /3<T<\delta /2\), and our next goal is to prove that U can be extended to a bounded solution of (3.13) on the whole set \(\mathbb {D}\). Clearly, it is sufficient to prove an a priori \(L^{\infty }\) bound of the solution on this domain in order to achieve the result via a continuation argument.
To this end we prove an energy estimate similar to the previous one, but now we integrate identity (3.15) over the slice
where \( \delta /3\le T_{1}<T_{2}<\pi \) are fixed, and we denote by F(T) the energy
After integration of (3.15), the terms on the side \(\alpha =\pi -T+\delta /4\) give a negative contribution at the RHS which can be dropped, as in the standard energy estimate, since the speed of propagation is 1. Proceeding as before, we are left with the inequality
and again the RHS here is negative thanks to (3.11). We conclude that the energy F(T) is nonincreasing:
Since \(F(\delta /3)\le E(\delta /3)\), by (3.18) we conclude
To proceed, we need a Lemma:
Lemma 3.5
Let \(I \subseteq (0,+\infty )\) be a bounded interval and \(n\ge 3\). Then for any \(V\in H^{1}(I)\) we have
with a constant independent of I and V.
Proof of the Lemma
Pick any points \(\alpha ,\beta \in I\) with \(\beta \ge \alpha /3\). We first prove that
We have two cases: either \(\alpha \le \beta \) or \(\alpha \ge \beta \ge \alpha /3\). If \(\alpha \le \beta \), we write
Using the inequality \(\int _{\alpha }^{\beta }s^{1-n}ds\le \alpha ^{2-n}\) and recalling that \(\alpha \le \beta \) we obtain (3.21). If \(\alpha \ge \beta \ge \alpha /3\), we have in a similar way
Now \(\int _{\beta }^{\alpha }s^{1-n}ds\le \beta ^{2-n}\) and we get
and recalling that \(\beta \ge \alpha /3\) we obtain again (3.21).
Next, split I in thirds \(I=I_{1}\cup I_{2}\cup I_{3}\) (with \(I_{1}\) at the left and \(I_{3}\) at the right). If \(\alpha \in I_{1}\cup I_{2}\), pick \(\beta \in I_{3}\) arbitrary and apply (3.21) to get
Since
(indeed \(\int _{a}^{b}sds=\frac{b^{2}-a^{2}}{2}\ge \frac{(b-a)^{2}}{2}\)) our claim (3.20) is proved for the points in \(I_{1}\cup I_{2}\). On the other hand, if \(\alpha \in I_{3}\), we pick \(\beta \in I_{2}\) arbitrary, we apply (3.21), and we get
and the same argument gives again (3.20). \(\square \)
We apply (3.20) to \(U(T,\alpha )\) at a fixed \(T\in (\delta /3,\pi )\) on the interval
note that \(|I|\ge \delta /4\). We get
For \(\alpha \in [0,\pi -\frac{\delta }{12}]\) we have
thus substituting in the previous inequality we get, for all \(\alpha \in [\Gamma (T),\pi -T+\delta /4]\),
Recalling the definition of F(T) and (3.19) we conclude
We now convert (3.22) into an estimate for u(t, x). Since \(\sin \alpha =\omega r\), we obtain
On the other hand, a change of variable shows that
where \(\widetilde{\mathbb {S}^{n}}\) denotes the image of \(\{t=0\}\times \Omega \) via the Penrose transform,
and
This gives the estimate
and recalling (3.22) (and the dependence of \(\delta \) in (3.16)), we conclude the proof of Theorem 1.3.
3.3 Proof of Theorem 1.4
Let u be the solution given by Proposition 1.2 and Theorem 1.3, and v be the local solution given by Lemma 3.4, maximally extended to a lifespan \([0,T^{*})\). Since the data are radial, v is also radial, and by the uniqueness part of Proposition 1.2 we see that \(u \equiv v\). In particular, v is bounded as \(t \uparrow T^{*}\), hence \(T^{*}=+\infty \). This proves the regularity of the radial solution.
It remains to prove the uniform bounds (1.11). For the \(L^{2}\) norm we have
Then we can write, if \(n\ge 3\),
by Hölder and Sobolev embedding, and we note the consequence of (1.9) (also valid only if \(n\ge 3\))
and the conservation of energy (1.5). Summing up, we obtain
where the integral converges since \(p>1+\frac{n+2}{n-2}=\frac{2n}{n-2}\). Noting that
we get the uniform bound
Estimate (1.11) for \(k=1\) is a consequence of the energy conservation \(E(t)=E(0)\) and of (3.25). For \(k>1\) we proceed by induction on k. By the energy estimate (2.10) we have
We apply Gagliardo–Nirenberg estimates to estimate the last term. Handling separately the highest order terms \(\sim \Vert u\Vert _{L^{\infty }}^{p-1}\sum _{j\le k} \Vert \partial ^{j}_{t}u\Vert _{H^{k-j}}\), we get
provided \(p>k+1\). Since u is radial in x we have
which is bounded by the induction hypothesis; on the other hand \(\Vert u\Vert _{L^{\infty }}^{p-k}\) is integrable since \(p>k+1\) by (3.24). Thus the right hand side is finite and this proves (1.11).
4 Global quasiradial solutions
This section is devoted to the proof of Theorem 1.5.
Let u be the global radial solution with initial data \((u_{0},u_{1})\) given by Proposition 1.2 and Theorem 1.4, and let v be the local solution with data \((v_{0},v_{1})\) given by Lemma 3.4. Denote by \(w=v-u\) the difference of the two solutions, which satisfies the equation
We can write
where
so that w satisfies
For \(j\le p-3\) we can write
where \(j_{1}+\dots +j_{\mu }=j\). By Sobolev embedding and energy estimates (1.11) we have
provided \(N>n+ \frac{n}{2} +j\), and this implies, recalling (3.24),
provided
Now let \(m\ge 1\) to be chosen and assume \(u_{0},u_{1}\) satisfy (4.5) with
while \(w_{0},w_{1}\) satisfy the compatibility conditions of order m. We see that if \(p>m+4\) the assumptions of Proposition 2.5 are satisfied and the estimates (2.22), (2.23) are valid.
Consider now the problem
where \(w_{j}=v_{j}-u_{j}\), \(j=0,1\), and \(V(t,x)=p|u|^{p-1}\). Define the spacetime norm
where the couple (q, r), satisfying (2.14), will be made precise below. Our goal is to prove the a priori estimate
where \(\epsilon \) is a suitable norm of the data \((w_{0},w_{1})\).
Applying (2.22), (2.23) to (4.6) we get
We must estimate the last term. We begin by noticing that
and if we assume \(m+1>\frac{n}{2}\) so that \(\Vert w\Vert _{L^{\infty }}\lesssim \Vert w\Vert _{H^{m+1}}\), we get
for \(0\le t\le T\). If we assume in addition \(m>\frac{n}{r}\) and use the embedding \(\Vert w\Vert _{L^{\infty }}\lesssim \Vert w\Vert _{W^{m,r}}\), we can write
provided \(p>q+2\). Since \(|u(t,\cdot )|\lesssim \langle t \rangle ^{-1}\) (recall (3.24)), we obtain
and integrating in time we conclude
We next estimate
Recalling the expression of F in (4.2), we may expand the derivative as a finite sum
where \(0\le \nu \le m\) and, assuming \(p\ge m+2\),
-
\(W_{1}=\partial ^{h_{1}}_{t}\partial ^{\alpha _{1}}_{x}w\) and \(W_{2}=\partial ^{h_{2}}_{t}\partial ^{\alpha _{2}}_{x}w\)
-
\(U_{k}= \partial ^{j_{k}}_{t}\partial ^{\beta _{k}}_{x}(u+\sigma w)\)
-
\(h_{1}+h_{2}+j_{1}+\dots +j_{\nu }=j\), \(\alpha _{1}+\alpha _{2}+\beta _{1}+\dots +\beta _{\nu }=\alpha \), and \(j+|\alpha |\le m\)
-
it is not restrictive to assume that \(j_{\nu }+|\beta _{\nu }|\ge j_{k}+|\beta _{k}|\) for all k and that \(h_{2}+|\alpha _{2}|\ge h_{1}+|\alpha _{1}|\)
-
G satisfies \(|G|\lesssim |u+\sigma w|^{p-\nu -2}\) so that \(\Vert G\Vert _{L^{\infty }}\lesssim (\Vert u\Vert _{L^{\infty }}+\Vert w\Vert _{L^{\infty }})^{p-\nu -2}\).
We take the \(L^{2}\) norm in x of each product, and we estimate it with the \(L^{2}\) norm of the derivative of highest order, times the \(L^{\infty }\) norm of the remaining factors; it may happen that the highest order derivative falls on w or on \(u+\sigma w\). Thus we need to distinguish two cases:
(1) The highest order derivative is on \(u+\sigma w\), that is to say \(j_{\nu }+|\beta _{\nu }|\ge h_{2}+|\alpha _{2}|\). Then we estimate
We have for \(0\le t\le T\)
Moreover for \(i<\nu \) we have \(j_{i}+|\alpha _{i}|\le \frac{m}{2}\), hence if we assume \(m>n-2\) we have by Sobolev embedding
where in the last step we used the energy estimates (1.11) for u, with
In a similar way
Finally, by (3.24) and Sobolev embedding
provided
Summing up we have, for \(m>\max \{n-2,n/r\}\), \(p\ge m+q+2\), \(0\le t\le T\)
We now integrate in t on [0, T] to get
with an implicit constant depending on \(\Vert (u_{0},u_{1})\Vert _{M}\), \(\Vert u_{0}\Vert _{H^{m+1}}\), \(\Vert u_{1}\Vert _{H^{m}}\).
(2) The highest order derivative is on w, that is to say \(j_{\nu }+|\beta _{\nu }|\le h_{2}+|\alpha _{2}|\). In this case we estimate
Proceeding in a similar way, we obtain again (4.10).
Summing up, recalling (4.8), we have proved
The implicit constant depends on \(\Vert (u_{0},u_{1})\Vert _{M}+\Vert u_{0}\Vert _{H^{m+1}}+\Vert u_{1}\Vert _{H^{m}}\). The conditions on the parameters are
and (q, r) satisfy (2.14), that is to say
All the constraints are satisfied if
with \(\delta <1\) chosen accordingly. Recall also that in order to apply Proposition 2.5 we assumed \(p>m+4=n+3\) and that \(u_{0},u_{1}\) are in \(H^{N}\times H^{N-1}\) with compatibility conditions of order N, where \(N>m+\frac{3}{2}n=\frac{5}{2}n-1\).
To conclude the proof it is then sufficient to apply a standard continuation argument; if the norm
of the initial data is sufficiently small with respect to the hidden constant
then the quantity M(T) must remain finite as \(T\rightarrow +\infty \) and global existence is proved.
5 Weak–strong uniqueness
Following [12], we prove a more general stability result which implies Theorem 1.6 as a special case. We use the notation
for the energy of a solution u(t, x) at time t of the Cauchy problem
Theorem 5.1
Let I be an open interval containing [0, T], \(T>0\). Let u, v be two distributional solutions to (5.1) on \(I \times \Omega \) such that
Assume in addition that v satisfies an energy inequality
Then the difference \(w=v-u\) satisfies the energy estimate
where C is a constant depending on
Proof
We shall need the following easy estimate of the \(L^{2}\) norm of \(w=v-u\):
The difference \(w=v-u\) satisfies in the sense of distributions
We expand
where
From the energy inequality for v, and the conservation of energy for the smooth solution u, we have
We get easily
which implies
Recalling (5.3), this gives
for some \(C=C(T,\Vert u\Vert _{L^{\infty }([0,T]\times \Omega )})\). On the other hand
which implies
and in conclusion
with \(C=C(p,T,\Vert u\Vert _{L^{\infty }})\).
In order to estimate B(t), we need the following remark. Assume the function
satisfies in \(\mathscr {D}'((0,T)\times \Omega )\), for some \(q\in (1,\infty )\),
Then for all \(\chi (t)\in C^{\infty }_{c}((0,T))\) and \(U(t,x)\in C^{\infty }_{c}(\mathbb {R}\times \Omega )\) we have the identity
Now assume
on an open interval \(I \supset [0,T]\), so that \(\square U\in C(I;L^{2}(\Omega ))\) and \(U_{t}\in L^{q'}(I\times \Omega )\). We can approximate U with a sequence \(U_{\epsilon }\in C^{\infty }_{c}(I \times \Omega )\) in such a way that
and
as \(\epsilon \rightarrow 0\) (e.g., extend U as 0 on \(\mathbb {R}^{n}\setminus \Omega \), apply a radial change of variables \(\widetilde{u}(t,x)=u(t,(1-\epsilon )x)\), truncate with a smooth cutoff \(\psi (|x|-\epsilon ^{-1})\) where \(\psi =1\) for \(|x|\le 1\) and \(\psi =0\) for \(|x|\ge 2\), and finally convolve with a delta sequence of the form \(\rho _{\epsilon }(x)\sigma _{\epsilon }(t)\)). Applying (5.8) to \(U_{\epsilon }\) and letting \(\epsilon \rightarrow 0\) we obtain that (5.8) holds for any functions U, W satisfying (5.6), (5.7), (5.9) and any \(\chi \in C^{\infty }_{c}((0,T))\).
Now, consider a sequence of test functions \(\chi _{k}(t)\in C^{\infty }_{c}((0,T))\), non negative, such that \(\chi _{k}\uparrow \mathbf {1}_{[0,t]}\) pointwise, \(t\in (0,T]\). We can write
Applying formula (5.8) with \(U=u\), \(W=w\), \(F=|u|^{p-1}u-|u+v|^{p-1}(u+w)\) (with \(q=(p+1)/p\)) and using the equations for u, w we obtain the identity
We have
which implies
Thus we get
Using (5.3) we conclude
for some C as in (5.2). Plugging (5.5), (5.2) in (5.4) we arrive at
with C as in (5.2), and by Gronwall’s Lemma we conclude the proof. \(\square \)
Data availibility
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Burq, N.: Global Strichartz estimates for nontrapping geometries: about an article by H. F. Smith and C. D. Sogge. Commun. Partial Differ. Equ. 9–10, 1675–1683 (2003)
Christodoulou, D.: Global solutions of nonlinear hyperbolic equations for small initial data. Commun. Pure Appl. Math. 39(2), 267–282 (1986)
D’Ancona, P., Lucà, R.: Stein–Weiss and Caffarelli–Kohn–Nirenberg inequalities with angular integrability. J. Math. Anal. Appl. 388(2), 1061–1079 (2012)
Ginibre, J., Velo, G.: Regularity of solutions of critical and subcritical nonlinear wave equations. Nonlinear Anal. 22(1), 1–19 (1994)
Kapitanski, L.: Global and unique weak solutions of nonlinear wave equations. Math. Res. Lett. 1(2), 211–223 (1994)
Metcalfe, J.L.: Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle. Trans. Am. Math. Soc. 12, 4839–4855 (2004). (electronic)
Shatah, J., Struwe, M.: Regularity results for nonlinear wave equations. Ann. Math. (2) 138(3), 503–518 (1993)
Shatah, J., Struwe, M.: Well-posedness in the energy space for semilinear wave equations with critical growth. Int. Math. Res. Not., (7):303ff., approx. 7 pp. (electronic), (1994)
Shatah, J., Struwe, M.: Geometric Wave Equations. Courant Lecture Notes in Mathematics, vol. 2. New York University Courant Institute of Mathematical Sciences, New York (1998)
Shibata, Y., Tsutsumi, Y.: On a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain. Math. Z. 191(2), 165–199 (1986)
Smith, H.F., Sogge, C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Commun. Partial Differ. Equ. 11–12, 2171–2183 (2000)
Struwe, M.: On uniqueness and stability for supercritical nonlinear wave and Schrödinger equations. Int. Math. Res. Not. Art. ID 76737, 14 (2006)
Conflict of interest
The author declare that they have no conflict of interest.
Funding
Open access funding provided by Universitá degli Studi di Roma La Sapienza within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is partially supported by the Project Ricerca Scientifica Sapienza 2016: “Metodi di Analisi Reale e Armonica per problemi stazionari ed evolutivi”
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
D’Ancona, P. On the supercritical defocusing NLW outside a ball. Anal.Math.Phys. 11, 141 (2021). https://doi.org/10.1007/s13324-021-00576-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-021-00576-3